WASHINGTON — Republican and Democratic senators clashed Tuesday on how best to overhaul Fannie Mae and Freddie Mac after a Treasury Department report on the troubled housing giants.

At issue are three long-term scenarios that Treasury posed for consideration in February for how to revamp the two government-controlled agencies, which were driven to the brink of collapse during the financial crisis in 2008.

The lack of consensus indicates that lawmakers have a long way to go before approving a statute, even though the Obama administration is urging Congress to approve a law within two years. The Senate Banking Committee, which is expected to draft legislation on the issue, on Tuesday held its first hearing on the housing industry since the Treasury released its report.

Republicans called for a virtually privatized system in which the government’s role in the mortgage market would be significantly reduced. In this scenario, only the Federal Housing Administration would be allowed to guarantee a small group of mortgages for low- and moderate-income borrowers who meet creditworthiness criteria.

“The less involvement by the public sector in housing, the better,” Sen. Bob Corker, R-Tenn., said in an interview after the hearing.

Democrats backed a broader guarantee approach, where the government would offer reinsurance for securities of a targeted range of mortgages.

Committee Chairman Tim Johnson, D-S.D., asked how the proposals would affect the ability of smaller banks to lend to underserved communities.

Shaun Donovan, who heads the Department of Housing and Urban Development, said a limited-guarantee program focused on FHA and Veterans Administration loans — as advocated by some GOP lawmakers — would likely lead to further consolidation, potentially hurting smaller banks that focus their energy on underserved communities.

“We have seen, with the effects of the crisis, significant consolidation in mortgage originations,” with the five largest lenders providing 60 percent of new mortgages, Donovan said.

Fannie and Freddie have long been crucial to the housing market, guaranteeing or owning roughly 50 percent of the residential mortgage market. Including FHA loans, the government has been responsible for about 95 percent of mortgage origination between 2008 and 2010, according to a Bank of America analysis.